Related papers: Double Affine Bruhat Order

Let $W$ be an irreducible Weyl group and $W_a$ its affine Weyl group. In a previous work the author introduced an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. The…

Given two affine permutations, some results of Lascoux and Deodhar, and independently Jacon-Lecouvey, allow to decide if they are comparable for the strong Bruhat order. These permutations are associated with tuples of core partitions, and…

We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield…

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of…

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their…

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on…

When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is…

The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering…

Let $F$ be a discretely valued complete field with valuation ring $\mathcal{O}_F$ and perfect residue field $k$ of cohomological dimension $\leq 1$. In this paper, we generalize the Bruhat decomposition in Bruhat and Tits from the case of…

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W.…

A diagram obtained from the Carter diagram $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate…

We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a…

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let…

Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…

We introduce a new language to describe the geometry of affine Deligne-Lusztig varieties in affine flag varieties. This first part of a two paper series develops the definition and fundamental properties of the double Bruhat graph by…

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy…

A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for…